The maximal free rational quotient

On Rational Points of Varieties over Local Fields having a Model with Tame Quotient Singularities

We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak Néron model of the base change of X to a tame Galois extension of K, then we construct a canonical weak Néron model of X with a map to this model, and examine its special fiber. As an application we get examples of singular models of X such that X has K-rational points specializing to a singular point of this model. Moreover we obtain formulas for the motivic Serre inv…

Picard 数为一且具有 $\mathbb{K}^*$ 作用、轨道商为 $\mathbb{P}_2$ 的 Gorenstein Fano 三维簇

Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$

本文分类了一类具有 $\mathbb{K}^*$ 作用和最大轨道商 $\mathbb{P}_2$ 的非环、$\mathbb{Q}$-因子、Gorenstein、Picard 数为一的 Fano 三维簇。

Quotients by actions of the derived group of a maximal unipotent subgroup

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. If $X$ is an affine $G$-variety, then the algebra of $U'$-invariants, $k[X]^U'$, is finitely generated and the quotient morphism $π: X \to X//U'$ is well-defined. In this article, we study properties of such quotient morphisms, e.g. the property that all the fibres of $π$ are equidimensional. We also establish an analogue of the Hilbert-Mumford criterion for the null-cones with respect to $U'$-invariants.

1-rational singularities and quotients by reductive groups

We prove that good quotients of algebraic varieties with 1-rational singularities also have 1-rational singularities. This refines a result of Boutot on rational singularities of good quotients.

极大幂幺有限商、法诺三维簇中的异常挠率，以及例外恩里克斯曲面

The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

本研究引入正特征下代数群概形的极大幂幺有限商，发现其应用于皮卡概形可编码异常挠率，借助例外恩里克斯曲面理论构造出存在该异常挠率的整法诺三维簇，打破了德尔佩佐曲面挠率为零的常规认知。

Quotients of cubic surfaces

Let $\Bbbk$ be any field of characteristic zero, $X$ be a cubic surface in $\mathbb{P}^3_{\Bbbk}$ and $G$ be a group acting on $X$. We show that if $X(\Bbbk) \ne \varnothing$ and $G$ is not trivial and not a group of order $3$ acting in a special way then the quotient surface $X / G$ is rational over $\Bbbk$. For the group $G$ of order $3$ we construct examples of both rational and nonrational quotients of both rational and nonrational $G$-minimal cubic surfaces over $\Bbbk$.

Quotients of del Pezzo surfaces of high degree

In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field $\Bbbk$ of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface $X$ contains a point defined over the ground field and the degree of $X$ is at least five then the quotient is always $\Bbbk$-rational. If the degree of $X$ is equal to four then the quotient can be non-$\Bbbk$-rational only if the order of the group is $1$, $2$ or $4$. For these groups we construct examples of non-$\Bbbk$-rational quotients.

On the Łojasiewicz exponent, special direction and maximal polar quotient

For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular $λ\in\bbC\{X,Y\}$ such that the Łojasiewicz exponent of $\grad\,f$ is not attained on the polar curve $\bJ(λ,f)=0$. When $f$ is not Morse we prove that for the same $λ$'s the maximal polar quotient $q_0(f,λ)$ is strictly less than its generic value $q_0(f)$. Our main tool is the Eggers tree of singularity constructed as a decorated graph of relations between balls in the space of branches defined by using a logarithmic distance.

Rationality of quotients by finite Heisenberg groups

We prove rationality of the quotient $\mathbb{C}^n / H_n$ for the finite Heisenberg group $H_n$, any $n \ge 1$, acting on $\mathbb{C}^n$ via its irreducible representation.

Towards an Intersection Chow Cohomology Theory for GIT Quotients

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…

$G$-constellations and the maximal resolution of a quotient surface singularity

For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_θ$ of $G$-constellations. It depends on the stability parameter $θ$ and if $θ$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_θ$ for some generic $θ$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.

具有小指数商奇点的有理同调射影平面研究

On rational homology projective planes with quotient singularities of small indices

本文研究拓扑与光滑阻碍对带小指数商奇点的有理同调复射影平面存在性的影响，完成了指数≤3且光滑轨迹一阶整同调平凡的这类曲面的商奇点类型分类。

A Hilbert-Mumford criterion for SL_2-actions

Let the special linear group $G := SL_{2}$ act regularly on a $Q$-factorial variety $X$. Consider a maximal torus $T \subset G$ and its normalizer $N \subset G$. We prove: If $U \subset X$ is a maximal open $N$-invariant subset admitting a good quotient $U \to U // N$ with a divisorial quotient space, then the intersection $W(U)$ of all translates $g \dot U$ is open in $X$ and admits a good quotient $W(U) \to W(U) // G$ with a divisorial quotient space. Conversely, we obtain that every maximal open $G$-invariant subset $W \subset X$ admitting a good quotient $W \to W // G$ with a divisorial qu…

Quotients of conic bundles

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over projective line by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group of order $2^k$, dihedral group of order $2^k$, alternating group of degree $4$, symmetric group of degree $4$ or alternating group of degree $5$ effectively acting on the base of conic bundle. Also we construct infinitely many examples of such quotients which are…

On maximal curves that are not quotients of the Hermitian curve

For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the Hermitian curve and raised the same question for $\mathcal X_\ell$ with $\ell>3$; in this paper we show that $\mathcal X_\ell$ is not Galois covered by the Hermitian curve for any $\ell>3$. Analogously, Duursma and Mak proved that the generalized GK curve $\mathcal C_{\ell^n}$ over $\mathbb{F}_{\ell^{2n}}$ is not a quotient of the Hermitian curve for $\ell>2$ and…

商叠的几何K-理论

The geometric K-theory of quotient stacks

本文将几何K-理论的构造精化至任意excellent基上商叠的有理K-理论，证明其是该叠K-理论内蕴分解的一部分，并建立了便于计算的诸多性质。

Cartier and Weil Divisors on Varieties with Quotient Singularities

The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.

Rationality of Three-Dimensional Quotients by Monomial Actions

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field $K(x,y,z)^G$ is rational (= purely transcendental) over $K$. Applications of this theorem will be given.

Torus quotient of Richardson varieties in Orthogonal and Symplectic Grassmannians

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.

Descent of line bundles to GIT quotients of flag varieties by maximal torus

Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.

Rational singularities and quotients by holomorphic group actions

We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …

有限群商空间的无分歧布劳尔群

Unramified Brauer group of quotient spaces by finite groups

本文提出了一个通用算法，用于计算有理簇在有限群作用下的商空间的无分歧布劳尔群。

楚 quotient 的两步方法

A two-step approach to Chow quotients

本文提出通过投影托里克品种和其有理数自同构群来编码楚 quotient 的几何结构，并应用于某些特定的有理数 homogeneous 变形。

Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles …

Hodge-Witt cohomology and Witt-rational singularities

We prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme…

Reconstructing function fields from rational quotients of mod-$\ell$ Galois groups

In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field $K$ of transcendence degree $\geq 5$ over an algebraically closed field, up-to inseparable extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed with the collection of mod-$\ell$ rational quotients.

Configurations of linear subspaces and rational invariants

We construct a birational equivalence between certain quotients of s-tuples of equidimensional linear subspaces of $C^n$ and some quotients of products of square matrices modulo diagonal conjugations. In particular, we prove the rationality of the quotient space of s-tuples of linear 2-planes in $C^n$ modulo the diagonal $\gl_n(C)$-action . Furthermore, we compute generators of the field of the rational invariants explicitly.

Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.

Fundamental group of a geometric invariant theoretic quotient

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line bundle on $M$. Assume that the GIT quotient $M/\!\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $π_1(M)\, \longrightarrow\, π_1(M/\!\!/G)$, induced by the rational map $M\, \longrightarrow\, M/\!\!/G$, is an isomorphism. If $k\,=\, \mathbb C$, then we show that the above rational map $M\, \longrightarrow \, M/\!\!/G$ induc…

Conformal blocks and rational normal curves

We prove that the Chow quotient parametrizing configurations of n points in $\mathbb{P}^d$ which generically lie on a rational normal curve is isomorphic to $\overline{M}_{0,n}$, generalizing the well-known $d = 1$ result of Kapranov. In particular, $\overline{M}_{0,n}$ admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of $\overline{M}_{0,n}$ as a conformal blocks line bundle. A symmetry in conformal…

On the growth behaviour of Hironaka quotients

We consider a finite analytic morphism $ϕ= (f,g) : (X,p)\to (\C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $π: (Y,E_{Y})\to (X,p)$ be a good resolution of $ϕ$ with exceptional divisor $E_{Y}=π^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $π$. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected compone…

Rationally connected varieties over the maximally unramified extension of p-adic fields

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ …

On the classification problem for the genera of quotients of the Hermitian curve

In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a self-polar triangle, or $q$ is even and $G$ is contained in the maximal subgroup $\mathcal{M}_2$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated to $\mathcal{H}_q(\mathbb{F}_{q^2})$. In this way several new values for the genus of a maximal curve over a finite field are obtaine…

A Closer Look ar Mirrors and Quotients of Calabi-Yau Threefolds

Let X be the toric variety (P^1)^4 associated with its four-dimensional polytope. Consider a resolution of the singular Fano variety associated with the dual polytope of X. Generically, anti-canonical sections Y of X and anticanonical sections of the resolution are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y,G) in X is not a quotient associated to an admissible pair in the resolution. Nevertheless, it is possible to construct a mirror orbifold of Z by means of a quotient of a …

有理极化Hodge结构的Mumford--Tate群的分类

Classification of the Mumford--Tate Groups of Rational Polarizable Hodge Structures

本文通过分解与极化有理Hodge结构相伴的pro-代数群G，将其商、导出群及伴随群分别刻画为Serre群、单连通覆盖及特定单代数群的乘积，从而分类了Mumford--Tate群。

Quotients of K3 Surfaces Modulo Involutions

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).

Rationally connected foliations on surfaces

In this short note we study foliations with rationally connected leaves on surfaces. Our main result is that on surfaces there exists a polarisation such that the Harder-Narasimhan filtration of the tangent bundle with respect to this polarisation yields the maximal rationally quotient of the surface.

Orbifold slope-rational connectedness

We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope Rational connectedness', and of orbifold `slope Rational quotient' . These notions extend to this larger context the classical notions of rationally connected manifold and `rational quotient' (sometimes called `MRC fibration'). Our notions and proofs work entirely in characteristic zero, and are based on the consideration of foliations with minimal positive slope with respect to some suitable movable class. The existence of covering or connecting families of `orbifold rational curves' is indeed presently unknown in the o…

A general Hilbert-Mumford Criterion

We provide a Hilbert-Mumford Criterion for actions of reductive groups $G$ on $Q$-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by $G$ from certain maximal open subsets admitting a good quotient by a maximal torus of $G$. As an application, we show how to obtain all invariant open subsets with good quotient for a given $G$-action on a complete $Q$-factorial toric variety.

Calabi-Yau threefolds of quotient type

First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample if the second Chern class is identically zero. We also derive a sufficient condition for the fundamental group to be finite in terms of the Picard number in an optimal form. Next, we give a concrete structure Theorem concerning $c_{2}$-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connec…

Uniformly rational varieties with torus action

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.

有理齐次簇中的极大不相交 Schubert 环

Maximal disjoint Schubert cycles in rational homogeneous varieties

本文通过研究经典类型有理齐次簇的 Chow 环中低余维数有效零因子及有效良好可除性，探讨了这些簇之间非常数映射的存在性，推广了射影空间和 Grassmann 簇的已知结果。

作为曲线乘积之商的有理正规曲面的循环覆盖

Cyclic coverings of rational normal surfaces which are quotients of a product of curves

本文研究作为曲线乘积之商的有理正规曲面的循环覆盖，推广 Esnault-Viehweg 方法证明单值群在第一 Betti 群及 Hodge 结构上的作用可分解为 P^1 上特定循环覆盖的直和，并应用于 Lê-Yomdin 曲面奇点和等平凡纤维化曲面。

Kähler几何中的超越数值维与有理商

Transcendental Numerical Dimension and Rational Quotients in Kähler Geometry

该研究探究紧Kähler流形的超越数值维性质，建立其与纤维化退化除子的联系，结合Cao-Păun定理回答有理商相关问题，得到伪有效线丛的数值维不等式。

Quotient curves of the GK curve

For every $q=l^3$ with $l$ a prime power greater than 2, the GK curve $X$ is an $F_{q^2}$-maximal curve that is not $F_{q^2}$-covered by any $F_{q^2}$-maximal Deligne-Lusztig curve. Interestingly, $X$ has a very large $F_{q^2}$-automorphism group with respect to its genus. In this paper we compute the genera of a large variety of curves that are Galois-covered by the GK curve, thus providing several new values in the spectrum of genera of $F_{q^2}$-maximal curves.

射影直线上动力系统的Chow商模空间与重标度极限

Chow Quotient Moduli Space of Dynamical Systems on the Projective Line and Rescaling Limits

本文研究射影直线上次数为d的有理自映射模空间的Chow商紧化，计算三维轨道环度数，引入带标记Chow商记录重标度极限，并扩展不动点乘子映射。

On the spectrum of genera of quotients of the Hermitian curve

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and group-theoretical description of $\mathcal M_q$ is given. The genera of some other quotients $\mathcal H_q/G$ with $G\not\leq\mathcal M_q$ are also computed. Thus we obtain new values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves. A p…

A New Triangulated Category for Rational Surface Singularities

In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.

分类曲线上极大矩阵序的布吕阿-蒂茨建筑的一个商

A quotient of the Bruhat-Tits building classifying maximal matrix orders over curves

本文将布吕阿-蒂茨树的商图构造推广到高维，研究曲线上极大矩阵序的商建筑，明确分裂极大序顶点的结构及与尖点区域的关联，为算术群和序的研究提供几何框架。

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Let G be a split simple group of type G_2 over a field k, and let g be its Lie algebra. Answering a question of Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g)^G. Soit G un groupe algébrique simple et déployé de type G_2 sur un corps k. Soit g son algèbre de Lie. On démontre que le corps des fonctions k(g) est transcendant pur sur le corps k(g)^G des invariants adjoints. Ceci répond par l'affirmative à une question posée par Colliot-Thélène, Kunyavskiĭ, Popov et…

复球商子簇的体积与有理点的稀疏性

Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points

本文证明复球商子簇体积受其 systole 控制，进而统一控制有理点增长率，并给出典范丛倍数的有效生成与非常丰沛性结果。

On maximal rational poylhedral fans

In this paper we study the geometry and combinatorics of the possible rational polyhedral fans with a given set of rays. The main questions we consider are when such fans are projective, complete, or simplicial. To answer these questions we use techniques of invariant theory to characterize the quotients of maximal saturated open sets in torus representations.

Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.